A myth is a female moth Student Bloopers

A myth is a female moth

Student Bloopers

When we think of Maths, we tendnot to think of Myths

Myths are the stuff of legend and wonder

Maths on the other hand, is coldly logical and has no room for doubt and error!

Or does it?

A mathematical truth, through retelling, an lack ofunderstanding, enters the public consciousness as a mythrather than a truth

Especially the case if this satisfies some underlying needfor some order and pattern in life, the universe, andeverything

A great shame as mathematical truths are far moreexciting and surprising than any myth, and give muchinsight into the way that the universe works

Myths about maths

Theseus and the Minotaur

Dido of Carthage

Iso-perimetric theorem

The scary maths myth

Mathematical myths

The Golden Ratio

The claim …

The Golden Ratio is a Divine Proportion

According to Mario Livio

Some of the greatest mathematical minds of all ages,from Pythagoras and Euclid in ancient Greece, throughthe medieval Italian mathematician Leonardo of Pisaand the Renaissance astronomer Johannes Kepler topresent-day scientific figures such as Oxford physicistRoger Penrose have spent endless hours over this simpleratio and its properties. ... Biologists, artists, musicians,historians, architects, psychologists, and even mysticshave pondered and debated the basis of its ubiquity andappeal. In fact, it is probably fair to say that the GoldenRatio has inspired thinkers of all disciplines like noother number in the history of mathematics.

The Truth about the Golden Ratio

Golden Rectangle

Link to the Fibonacci Sequence [Kepler]

1 1 2 3 5 8 13 21 34 55 89 144 …

1 2 1.5 1.6666 1.6 1.625 1.615 1.619 1.617 1.618 1.618 …

Ratio between successive terms

Rapidly approaches the Golden Ratio

nth Fibonacci number is given by

More Geometry

Ruler and compass construction of the Golden Ratio

Link to the Pentagon

The Pentagram and Penrose Tilings

Irrationality of the Golden Ratio

There are no integers m and n such that

So .. Does the the Golden Ratio play a ‘central role in mathematics’?

NO!!! This is a myth

As a number it is only in the Championship League

Numbers

Bottom of the Premier league

Top of the Premier league

4

Exponential growth

In my own work I use these two numbers all the time

I have used the Golden Ratio exactly twice

Practically every formula in science and engineering involves e, or pi or a combination of the two

Other great numbers

0, 1, -1, i

The most important formulae in mathematics

No sign of the Golden Ratio here!

Links of the Golden Ratio to Nature

The truth

The Golden Ratio appears in nature when there is five-foldsymmetry

Eg. Quasi crystals

However these crystals are rather uncommon compared to cubes and hexagons which involve the square roots of 2 and 3

The Golden Ratio is linked to the Fibonacci sequence

This arises naturally in studies of population growth

and also the way that objects pack together

Spirals in a sunflower

Drone bees in a bee hive

The myth

Much much more is claimed of the Golden Ratio

Supposed to be at the heart of many of the proportions in the human body.

Eg. ratio of the height of the navel to the height of the body

Just about every proportion of the perfect human face

However, none of this is true, not even remotely !!!

The human body has lots of ratios between 1 and 2

Some by chance are ‘close’ to the Golden Ratio

Many are not

Tendency of the human brain to look for patterns

Spirals, Golden and otherwise

Golden spiral …. It’s not a spiral!

Logarithmic Spiral

Because of … Self-similarity

If you rotate the spiral by any fixed angle then you get a spiral which is a rescaling of the original

Very common in nature

a and b can be any value!!

Golden Spiral

Nautilus:

The Golden Ratio in Art

The truth: Some artists eg. Le Corbusier have consciously used the Golden Ratio

The myth: The Golden Rectangle is supposed to be the most aesthetically pleasing rectangle

This has been tested. There is no evidence for it

Claimed da Vinci used the Golden Ratio in his art

No direct evidence of this!

da Vinci only mentioned whole number ratios

Famous example: Vitruvian ManBut its proportions do not match the Golden Ratio

Examples of finding the Golden Ratio in his pictures are in the same class as finding the ratio in the face

The Parthenon

No evidence of the use of the ratio in Greek scholarship

Idea the Parthenon has proportions given by the Golden Ratio only dates back to the 1850s

The real truth is much better than the myth

The Golden Ratio is the most irrational number

z any number m/n a fraction

error of approximating z by a fraction

21

34

55

89

13

8

5

Hard to approximate

Easy to approximate

The Monty Hall Problem

Contestants on the game show are shown three shut doorsBehind one is a car Behind the other two is a goat

Asked to choose a door and to tell the hostThe host opens a different door to reveal a goat

Given a choice

Stay with the chosen door or swap

The door they choose is opened

Should you change yourdoor or not?

Often quoted answer is YES!

But this isn’t always correct

Answer depends completely on the knowledge of the host and contestant

Case 1: Knowledgeable host, ignorant contestant

Chance contestant chooses a car is 1/3

Knowledgable host will always reveal the goat so no new information

Chance the other door is a car is still 2/3

Double your chances by changing doors

Case 2: Ignorant host, any contestant

Contestant chooses a door

Host shuts their eyes and opens a door. It’s a goat

Changes the information in the problem

Bayes’ theorem says that the chance of the contestant having a car is now ½

No reason to change

Case 3: Knowledgeable host, knowledgeablecontestant

A. Contestant chooses a car. Host reveals goat. Contestant changes their door. They get a goat!

B. Contestant chooses a goat. Host asks them to choose another door, and then choose whether to swap doors. They swap and get a car with probability ½

Overall they only get a car with probability 1/3

A little knowledge is a dangerous thing

Rob Eastaway

The Four Colour Theorem

Want to colour a map as cheaply as possible

Rules

1. Each country must haveone colour

2. Two countries whichhave a common border

must have different colours

Q. What is the smallest number of colours needed?

Must have at least four colours

Found ‘empirically’ that only four colours seemed to be needed

Conjecture proposed in 1852 by Francis Guthrie, who was trying to colour the map of counties of England

‘Proof’ in 1879 by Kempe

Shown to be wrong. But became the‘five colour theorem’

Resisted proof for 100 years

1976 the four colour theorem was proved by Kenneth Appel and Wolfgang Haken

Proof was ‘by computer’ and was Very controversial

Now a very important result in communications technology

The Truth

The four colour theorem applies to all non contiguous planar graphs

The Myth

It works for maps

It doesn’t!

A map which cannot be coloured with four colours

Lake Gresham

Lake Geometry

1

2

3

4

Problem: The sea is always blue

Empires can be red

Maps like this often need more than four colours

Cutting a cake

Fair cut Unfair cut

How do you cut a cake fairly?

The Myth: The I cut, you choose, method

One party divides up the cakeThe second party then has the first choice

Reasoning: it is clearly in the interests of the first party to cut the cake as fairly as possible

Then, no matter how the second party choses, the remaining piece will be as close to ½ of the original as it is possible to get

Problem:

This method gives an overwhelming advantage to the person who choses first

A much better method is to use iteration

This is how a computer would do it

First Cut

Second Cut

Continue with the new piece till only the crumbs are left

A mathematician named Hall,

Once went to a fancy dress ball,

They thought they would risk it,

And go as a biscuit,

But a dog ate them up, crumbs and all.